In fact it is often useful (such as in the symplecto-geometric interpretation that we turn to below) to restrict attention to non-vanishing solutions (or else to solutions restricted to their support) in which case we can regard ϕ\phi as a function of the form

where dS\mathbf{d} S is the gradientcovector field of SS, where d†(a2dS)\mathbf{d}^\dagger ( a^2 \mathbf{d}S) is the divergence of a2dSa ^2 \mathbf{d}S, and where 𝒪(ℏ2)\mathcal{O}(\hbar^2) denotes all further terms that are non-linear in ℏ\hbar.

This means that ψ(−,t)=exp(iS/ℏ)aexp(−iω)\psi(-,t) = \exp(i S / \hbar) a \exp(- i \omega ) is a semiclassical stationary state with energy

E≔ℏω
E \coloneqq \hbar \omega

if the phase SS and the modulus aa satisfy the following two conditions:

for the restriction of the cotangent bundle projection to this Lagrangian submanifold.

The fact that SS satisfies the Hamilton-Jacobi equation means equivalently that this Lagrangian submanifold is the level-set of the HamiltonianH:ℝn→ℝH \colon \mathbb{R}^n \to \mathbb{R} at energy E=ℏωE = \hbar \omega

im(dS)=H−1(E).
im(\mathbf{d}S) = H^{-1}(E)
\,.

For the interpretation of the modulus function aa in this reformulation, first notice that for volvol the canonical volume form on ℝn\mathbb{R}^n, the homogeneous transport equation